The authors in [1]} select a number \(K_4^{-1}\le \rho \le 1\)  and a set \(\bar{A}\subset A\) , with \(\#\bar{A}\gtrsim |\log \delta |^{-1}(\#A)\) , so that
\(\#\big ( (aA)_{2\delta ^+}\cap (bA)_{2\delta ^+}\big )\sim (\# A)K_4^{-1}\rho ^{-1}\quad \textrm {for each}\ a\in \bar{A}.\) 
